[QUOTE=paindoc;46679517]Okay so I'm not the biggest fan of general mathematics since I don't usually see their applications very well, physics is more of my thing. [/QUOTE]
This is a very common (and imo naive) viewpoint, but as you progress through physics, you will likely find every area of math you could possibly have seen applied somewhere in physics. Differential geometry, algebraic geometry, topology, and even category theory have applications in physics. I'm sure someone's applied Morse theory somewhere.
Also, here's a great quote by the late Sidney Coleman: “The career of a young theoretical physicist consists of treating the harmonic oscillator in ever-increasing levels of abstraction.” It's rather accurate, and as always in physics there is reason! The leading order approximation to a potential around a minimum (where systems like to settle) is a quadratic potential: the harmonic oscillator! I was thinking about this the other day whilst very bored at work.
[QUOTE=JohnnyMo1;46679952]This is a very common (and imo naive) viewpoint, but as you progress through physics, you will likely find every area of math you could possibly have seen applied somewhere in physics. Differential geometry, algebraic geometry, topology, and even category theory have applications in physics. I'm sure someone's applied Morse theory somewhere.
Also, here's a great quote by the late Sidney Coleman: “The career of a young theoretical physicist consists of treating the harmonic oscillator in ever-increasing levels of abstraction.” It's rather accurate, and as always in physics there is reason! The leading order approximation to a potential around a minimum (where systems like to settle) is a quadratic potential: the harmonic oscillator! I was thinking about this the other day whilst very bored at work.[/QUOTE]
When you think about it, literally everything in maths revolves around either adding, subtracting, multiplying and dividing, just in different steps and different ways. Or, graphs, which is the one thing that seems to be connected to literally every area of maths somehow, and and graphs are also present in physics. For example, let's say I had a really long v-t graph with a curve (I know this is basic but it's an easy example). Rather than awkwardly find the area under the curve to get the distance travelled, you could just integrate to find the area rather easily. Saves a lot of time, honestly :v (yes i know trapezium rule but you know you might want an exact result)
[QUOTE=JPlus;46686422]When you think about it, literally everything in maths revolves around either adding, subtracting, multiplying and dividing, just in different steps and different ways.[/quote]
I think this is a rather narrow way of looking at things.
[QUOTE=JPlus;46686422] Or, graphs, which is the one thing that seems to be connected to literally every area of maths somehow, and and graphs are also present in physics. For example, let's say I had a really long v-t graph with a curve (I know this is basic but it's an easy example). Rather than awkwardly find the area under the curve to get the distance travelled, you could just integrate to find the area rather easily. Saves a lot of time, honestly :v (yes i know trapezium rule but you know you might want an exact result)[/QUOTE]
This too, there's an awful lot of things for which graphs aren't very helpful.
[QUOTE=JPlus;46686422]When you think about it, literally everything in maths revolves around either adding, subtracting, multiplying and dividing, just in different steps and different ways.[/QUOTE]
Not... really...
[QUOTE=JPlus;46686422]When you think about it, literally everything in maths revolves around either adding, subtracting, multiplying and dividing, just in different steps and different ways.[/QUOTE]
Can you prove Poincaré's conjecture for me by just using the addition, subtraction, multiplication and division operators? I'll be nice and let you use graphs too if you want.
[QUOTE=JPlus;46686422]When you think about it, literally everything in maths revolves around either adding, subtracting, multiplying and dividing[/QUOTE]
I think you could say that a part of maths is abstracting the idea of those operations, but even then that's not close to "literally everything".
[QUOTE=agentalexandre;46688293]Can you prove Poincaré's conjecture for me by just using the addition, subtraction, multiplication and division operators? I'll be nice and let you use graphs too if you want.[/QUOTE]
Well the Poincare theorem is implied by the geometrization theorem, which is about connected sums of manifolds :>
Okay let me rewrite my earlier statement.
[B]Generally[/B] all you need for most fields at their core are adding, subtracting, multiplying and dividing, in specific ways and order, and with specific interpretation of any data you're given, which includes both real and imaginary numbers, letters, visual data, and any other variables. Also the ability to look at things and know that a + b = a + b. Generally.
Like something that seems like a long complicated formula is really just doing this with that then doing that and that. Like you differentiate, you interpret the data (i.e. basic differentiation problem of y=5x^4, you see the "4" and "5" and multiply to make the new coefficient, then subtract 1 from 4 to dy/dx=20x^3 (i'm trying to not bring up super long examples because it'll clog up the thread with shit that can be shown in a shorter detail)). Even proof, like algebraic, or by induction, is just using those same operators with variables, of course I say generally and should've said that because if you go to somewhere like decision maths you won't find as many of those there :v but I digress, I apologise, I should've said "generally" rather than "literally everything"
[QUOTE=Krinkels;46687496]This too, there's an awful lot of things for which graphs aren't very helpful.[/QUOTE]
I was providing an example of a use? I know graphs only represent data, and aren't that useful in a lot of fields, I just said they seem to be connected to a lot of things in maths and are present in physics.
[QUOTE=ThisIsTheOne;46688388]I think you could say that a part of maths is abstracting the idea of those operations, but even then that's not close to "literally everything".[/QUOTE]
Yeah that was what I was kinda getting at. Though I'm not sure if you can say there's a majority of fields in maths that don't use multiplication, division, addition or subtraction at any single point throughout the entireity of that field.
I mean I'm speaking on a basis from Further Maths and Physics at A-level so obviously I won't be as educated per se as some of you who could have for all I know degrees, so excuse me if I make a dumb error in what I say like that :P
So I fucked up pretty hardcore.
I got a 2.0 on my Calc I final and passed by literally 1.6%. I've never done this poorly in a class in my entire life, and I've never bombed a final that hard in my life either (39/100, still within 1 standard dev tho). I definitely didn't feel like I did that bad, and I did do 3 practice finals before the test but I ate shit anyways.
I've never really had to study for math before. With Physics, I have a book that goes over a lot of concepts and text and theories combined with a great prof that I can rely on to give good lectures and answer questions. Its the opposite for math. My book barely intros formulas and basic theories and then lists a bunch of problems with maybe a few solutions. Even the story problem selection is slim to nonexistent. My prof has a thick accent and didn't give very good examples, and didn't answer questions well.
Combine that with the fact I really only studied for the final and yeah, you get what I got. I messed up, how do I fix this? How should I study for a college math test? How do I best [I]learn[/I] what I need to know instead of just memorizing for a test?
It's only one 2.0 but I really don't intend to repeat my mistake ever again, this is one hell of a wake-up call. I'm losing sleep and if my parents find out (they're footing the bill, I owe them better performance) I'm turbofucked. They've said if I get below a 3.0 more than once financial support is out, and since my Dad is a doctor I'm fucked if he pulls his support (no need based scholarships for this guy). So I want to do better
thanks
[editline]12th December 2014[/editline]
sorry if this is the wrong thread, but it seemed like if you guys are delving deep into the field you probably have some pretty solid advice on how to study and do well in mathematics. I'm starting to like the field more as I see how it links to physics and all (and I enjoy solving problems), but performance like this and a prof like that saps all the fun out of it
You really have to be pro-active about things like this. If you're feeling uncertain about the content, ask the tutor, or go the Prof's contact hours. Ask them which sections of the text (if they prescribe one) are most useful, but also don't neglect alternative sources like Khan Academy or PatrickJMT. Failing that, form a study group/ask around on the course's Facebook page (if there is one); if you're having trouble with a topic, chances are someone else is too.
In regards to studying for the exam, I personally find that doing practice exams/examples isn't enough. I think it's much more valuable (and rewarding) to cement your understanding of the motivation and general principles of the concept. You never know, the exam question might be wildly different to any example you looked at - and if you don't understand the problem in the first place and were instead relying on recognizing the structure of an example question then you're pretty much doomed to misery and vice.
I never do practice exams because they make me stress the fuck out. I skim over them and nothing's immediately obvious so I'm like "Shit I don't know a thing", but then during the exam my mind goes into overdrive and it usually works out okay :v:
I also don't have time to do practice exams, I'll be glad if I can just finish the theory and get some exercises done.
Also what PopLot said, it might hamper your "out of the box" thinking. Professors can get creative when making up exams.
[QUOTE=Number-41;46697350]I never do practice exams because they make me stress the fuck out. I skim over them and nothing's immediately obvious so I'm like "Shit I don't know a thing", but then during the exam my mind goes into overdrive and it usually works out okay :v:
I also don't have time to do practice exams, I'll be glad if I can just finish the theory and get some exercises done.
Also what PopLot said, it might hamper your "out of the box" thinking. Professors can get creative when making up exams.[/QUOTE]
The opposite can also apply. They can throw something at you that's so simple if you know what you're talking about but is an intractable nightmare if you don't.
[QUOTE=Number-41;46697350]I never do practice exams because they make me stress the fuck out. I skim over them and nothing's immediately obvious so I'm like "Shit I don't know a thing", but then during the exam my mind goes into overdrive and it usually works out okay :v:
I also don't have time to do practice exams, I'll be glad if I can just finish the theory and get some exercises done.
Also what PopLot said, it might hamper your "out of the box" thinking. Professors can get creative when making up exams.[/QUOTE]
Honestly when I did my GCSEs I did practice papers and I got a high A* (though GCSEs are actually really easy but I digress). You don't have to start off practice papers making it seem like an exam environment, you can start with a text book and if you stumble on any questions, look up how to do them. Progress on until you can finish them without a textbook and in a set time, and you should be good. It doesn't only teach you how to do the question, per se, but how examiners throw in trick questions and what you can look out for. Spotting similarities in exams is brilliant because they come up almost all the time over here at least and it's basically a free headstart.
But at the same time, don't just do practice papers if you're not getting used to the concepts. As PopLot said, asking your teacher (and preferably your teacher rather than a tutor, because the teacher likely knows more about how you learn and what level you're at) or using sites/sources like Khan Academy is really good, and since the paper designers always find new ways to trick you in some way by either combining two different areas or expanding on a certain area, which the method for working out would have to be realised.
Also the smart people in this thread are here, if you ever need any of them, they likely have a much better understanding of all this than you or I do, so they're definitely worth a shout for help if you can't get it from anyone else :3 I wish you good luck next exam!
[QUOTE=PopLot;46697511]The opposite can also apply. They can throw something at you that's so simple if you know what you're talking about but is an intractable nightmare if you don't.[/QUOTE]
Also I've noticed a lot of the time they try to make questions seem a lot more complicated than they actually are, like throwing in random pieces of infomation that don't apply but are just there to confuse you (i see that in mechanics and also some core subjects too).
had an interview for acceptance into a Ph.D programme in medical imaging on tuesday, got an email today saying they were impressed and were willing to have me, and offered me a studentship! I have about a hundred different thoughts about this, a lot of excitement and some fear, but at least I now have a working plan for next year. if anyone has any general tips for this kind of thing, please tell me
Thanks for the advice guys. I'll do my best. I've calmed down a lot, its best I fuck up and learn how not to do it again my first quarter freshman year rather than later down the road when my schedule is even harder.
I'm gonna do better next semester, I know it
So I know mathematicians like to build up manifolds by starting by saying they're a topological space first. Physicists are too lazy to learn topology so they don't like to do that. The say it's a set with a collection of subsets that cover it, transition maps are diffeomorphisms etc., and then usually put in a footnote about being able to induce a topology on the manifold from the charts.
So I've seen two ways of doing this. One is to declare that all the chart maps are homeomorphisms. The other is to say that a subset of the manifold is open if and only if the image of its intersection with all chart domains under the associated chart map is open. It didn't seem obvious to me that these induce the same topology, and I can't find a proof of it anywhere, so I'm trying to prove it (now it does seem pretty obvious, this is why we prove things for ourselves!).
I think my proof is fine but I have one sticking point: assuming that the chart maps are homeomorphisms and trying to show that if the image of the intersections are open, the set is open. My thought was to say that, since the chart maps are homeomorphisms, all the intersections of the set with chart domains must be open, but my problem is that I don't know if that gives me that they're open in the manifold, or just in the chart domain as a subspace. I know that if the chart domains are open (and they are, I just don't know how to prove it) it's easy, because then the intersections are definitely open in the manifold, and I can union them all together to get that my set is open in the manifold.
[B]tl;dr[/B] Does a homeomorphism from a subspace of a topological space to an open subset of Euclidean space give me that the subspace is open in in the topological space? Alternatively, how do I show that chart domains in a manifold are open, demanding that the chart maps be homeomorphisms?
So I know mathematicians like to build up manifolds by starting by saying they're a topological space first. Physicists are too lazy to learn topology so they don't like to do that. The say it's a set with a collection of subsets that cover it, transition maps are diffeomorphisms etc., and then usually put in a footnote about being able to induce a topology on the manifold from the charts.
So I've seen two ways of doing this. One is to declare that all the chart maps are homeomorphisms. The other is to say that a subset of the manifold is open if and only if the image of its intersection with all chart domains under the associated chart map is open. It didn't seem obvious to me that these induce the same topology, and I can't find a proof of it anywhere, so I'm trying to prove it (now it does seem pretty obvious, this is why we prove things for ourselves!).
I think my proof is fine but I have one sticking point: assuming that the chart maps are homeomorphisms and trying to show that if the image of the intersections are open, the set is open. My thought was to say that, since the chart maps are homeomorphisms, all the intersections of the set with chart domains must be open, but my problem is that I don't know if that gives me that they're open in the manifold, or just in the chart domain as a subspace. I know that if the chart domains are open (and they are, I just don't know how to prove it) it's easy, because then the intersections are definitely open in the manifold, and I can union them all together to get that my set is open in the manifold.
[B]tl;dr[/B] Does a homeomorphism from a subspace of a topological space to an open subset of Euclidean space give me that the subspace is open in in the topological space? Alternatively, how do I show that chart domains in a manifold are open, demanding that the chart maps be homeomorphisms?
anybody know when you use a matrix and gauss jordan elimination to solve a crank-nicolson method and when you generate a grid to solve it
this is for a heat-transfer problem too
My guess is that you'll need GJ if it is an implicit method. Also generating a grid happens before you set up the matrix, as that matrix will act on a vector that is a discrete approximation of your continuous function. You basically make your differential equation a linear algebra problem (dU/dt=AU+g, where A represents your partial space derivatives and g encodes your boundary conditions).
That or I misread your question. [URL="http://www.dynamicearth.de/compgeo/Tutorial/Day2/cranknicholson.pdf"]This pdf[/URL] summed it up really nicely when I had to solve the Black Scholes equation numerically.
[QUOTE=Number-41;46713989]My guess is that you'll need GJ if it is an implicit method. Also generating a grid happens before you set up the matrix, as that matrix will act on a vector that is a discrete approximation of your continuous function. You basically make your differential equation a linear algebra problem (dU/dt=AU+g, where A represents your partial space derivatives and g encodes your boundary conditions).
That or I misread your question. [URL="http://www.dynamicearth.de/compgeo/Tutorial/Day2/cranknicholson.pdf"]This pdf[/URL] summed it up really nicely when I had to solve the Black Scholes equation numerically.[/QUOTE]
thanks, from the context i believe my problem its an explicit method, and just a quick search sort of confirms this.
[editline]14th December 2014[/editline]
ya this is what i'm doing currently, i still have to ask the professor some questions about this problem, but from the 6 or so other people from my class working on this, i appear to be the only one to actually get this far
Can anyone tell me how to enter an implicit differentiation function like
Find y' if x=tan(x+y) into a Ti-Nspire CAS?
I put impDif(tan(x+y)=x,x,y) but it's not giving me the answer I was hoping for
Should come down to -sin^2(x+y) but instead i'm getting
180*(cos(x+y))^2 - (pi)
------------------------------- <--- attempted fraction bar
(pi)
^is this the same thing possibly?
Not exactly math but it's math
Same thing. sin^2(x) + cos^2(x) = 1, and also some weird thing it's apparently doing with degrees and radians. I think. That looks like where the pi and 180 come from.
It's in degrees mode i'm pretty sure c:
Still learning to use this calculator
Could anyone give me a hand with this problem?
[t]http://puu.sh/dB6mK/3a72eee5c3.jpg[/t]
H,A and B are in a horizontal plane, I have to find the line HA in functions of the angles and the distance between A and B, wich is D. It's trigonometry, so the sinus and cosine theorems may come in handy.
Find the length of AM in terms of AB and alpha.
Find the length of AH in terms of AM and gamma.
Then it's some algebra/substitution.
Yep, I just did it that way, thanks!
Dunno if it was posted here already, but James Stewart died this month.
:(
So I was watching the MIT opencourseware class on mathematics for computer science. The professor gave the students a problem:
There is a proposed courtyard to be built, of dimensions 2^n * 2^n. There must be a statue in the (near)-middle for the guy who donated enough money to build the courtyard, bill. SO the courtyard, for example, would look like this for p(2) (2^2 * 2^2):
[img]http://i.imgur.com/0djGfQt.png[/img]
Where bill is the red, in the near-center.
The courtyard, however, must be constructed with 3-square L-shaped blocks, that look like:
[ ]
[ ][ ]
The professor eventually changed the theorem so that the statue could be anywhere, as they couldn't find a place in the center for it for all n. They did the problem by induction but I found what I thought was an easier method, and I wanted to see if anyone could check it:
so the theorem p(n) is that for all n >= 0, there exists a way to construct a courtyard using L-shaped blocks such that there will be any square left over for bill's statue.
the base case was that p(0) = 2^0 * 2^0, which is 1. The only block they had was bills statue, so that was ample proof that p(0) can be assumed.
What I did was to multiply 2^n by 2^n to get 2^2n, which is 4^n. So to find one block left over when constructed with blocks of 3 is really trying to prove that:
for all n >= 0, 4^n modulo 3 = 1, or that all powers of 4, when constructed by threes, will have a remainder of 1.
Isn't that still by induction?
[QUOTE=Krinkels;46766243]Isn't that still by induction?[/QUOTE]
Idk if it is but its not how they did it. I just wanted to know if I was correct
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